A resistor network with a curious property

Each resistor dissipates about 10% of the total, so with 1/4-watt resistors, the whole thing can be rated for about 5W. A 40dB 5W attenuator. Now just to source all those odd 0.001% values.

--
 Thanks, 
    - Win

No, it's a sensible size... but whatever value you want always seems to be out of stock.

NT

What are you solving for?

Bingo! I first evaluated this approach, and noticed the dissipation-per-resistor made a trio-repeating pattern: a suboptimal result. It also used a lot of resistors.

The observation that a ladder is alternating tee and pi sections is also a nice one. Finding the symmetries of networks is useful (yes, even if you're just going to fiddle with values and hand- or machine-optimize them). :)

Tim

--
Seven Transistor Labs, LLC 
Electrical Engineering Consultation and Contract Design 
Website: http://seventransistorlabs.com

Heh... for a chain of infinity, all the series resistors are lim-->0, and all the parallel resistors are lim-->infty.

Not a very practical infinite series, as serieses go. :^)

Which reminds me......

Hmm, curious why you'd think that. A ladder network is basically a continued fraction: a well studied, not-difficult-to-work-with, and strictly rational structure. Even if you went whole hog and wrote out the optimization problem itself (a multidimensional polynomial), it may be possible to simplify it to a closed form (rather than being stuck behind an intractible polynomial root, or behind an integral that converges but has no finite analytical form and therefore has to be computed). Do you suspect one of these possibilities?

One of the great developments in math, since the 19th century, is the trilemma of problem solving: either it's provably solvable (like x = 1), provably unsolvable (like x^2 = -1 over the set of real numbers), or provably unprovable.

And the latter has a number of fairly common examples: it could be conformal to one of the currently unknown problems (like the Riemann hypothesis), or it could be self-referential (Goedel incompleteness) or incomputable.

So the wonderful thing is, if it doesn't fit into one of these impossible options, and it doesn't involve any of the hard options, then there /must/ be a solution you can write down, and it's your own fault if you're too lazy to reach out and find it! :)

I think that's very encouraging! You can always have a positive conclusion (that it fits into one of these bins), even if it's not the solution you were initially hoping for.

Tim

--
Seven Transistor Labs, LLC 
Electrical Engineering Consultation and Contract Design 
Website: http://seventransistorlabs.com

In practice, quite good results would be had with 1% resistors (i.e., power matching to 2%, SWR < 1.01). I didn't bother truncating the decimals, it looks more impressive that way. ;-D

You might not even get 1% values in power resistors (say for a 100W+ version), necessitating some stacking of series and parallel values to get acceptable accuracy.

Likely, quite a good tradeoff (better than 10%) can be found just by playing around with the set of nearest E6 values.

I wonder how much effect the physical length of the resistor will have. At frequencies where the resistor itself is 1/4 wavelength long, the transmission line length has the effect of 1/4-wave-transformering half of the resistor from a series equivalent to a parallel equivalent, or vice versa. The effect is higher loss per resistor. The SWR should change slightly at very high frequencies, but more importantly, the power gets concentrated towards the beginning of the network, reducing its ultimate power limit. (Skin effect will behave similarly, too.)

Resistors with high impedances (aka inductive wirewound resistors) would also be problematic for the series resistors, but not necessarily a big issue for the early parallel resistors (which may have values on par with their AC impedances). As the series and parallel values both converge towards 50 ohms, the characteristic impedance should converge as well, making bulk film parts more desirable than wound or sinuous-path-etched parts.

Tim

--
Seven Transistor Labs, LLC 
Electrical Engineering Consultation and Contract Design 
Website: http://seventransistorlabs.com

The contrasting subjective experiences of "challenge" seen in this thread are interesting to me. Bitrex seems to think it's in some way intractible; I think it's no worse than quadratic (and some others seem to support that intuition); and you think it's a "challenge" (not for the given problem, which was solved to your satisfaction -- if it has numbers, it works -- but for a related problem, which is always fun to think about).

Yet for this related problem, I can instantly see why that cannot be. (I have the unfair advantage of having created and solved the problem in the first place. Not trying to rub it in, but I think I'd like to leave the answer to be puzzled out, so I won't show why, right now anyway.)

So I wonder if your sense of "challenge" is calibrated more along the lines of a manager -- it calls to mind some recent debacles from the crowdfunding world: their managers seem to well and truly believe that "we'll solve it in development", nevermind that what they're trying to solve happens to be thermodynamics itself!

Fortunately, you know better than to fight thermodynamics itself, but the fact remains: if it's not immediately obvious that the problem being solved is equivalent to thermodynamics, or causality, or something like that, then you want to continue pushing for that solution, as long as it might be possible.

It's not an indictment of your scientific knowledge, indeed it's a very practical, managery thing to do, to push for a solution regardless of the nature of the problem. I mean very specifically that it's a "calibration of 'challenge'". It's not practical for a manager to have complete knowledge of the subproblems going into their projects, it's only their duty to make sure they get solved. And, if it should happen that the problems cannot be solved, adjust the schedule or project design to avoid that pitfall.

I just think it's interesting to see different worldviews applied to the perception of a problem.

Hmm, metal thin film, perhaps? Or maybe a conductive plastic that happens to have that color. Fun in any case.

Looks like they did a pi attenuator, more or less, which limits power on the first resistor, probably. What is that, 1/2W or 1W, something around there?

Interesting that they have a bunch of unused pads on that substrate. I wonder if they use the same substrate, with different carbon tracks and bond-outs, for a whole slew of parts.

Close enough values are certainly available in spades; the resulting network would only be good to a watt or two, though. You could just as well make a couple low-dB attenuators (in the 1-10dB range) out of 1218s or 2512s, to get a couple watts out of chip resistors, without sacrificing much bandwidth.

My initial concept was to use thick film power resistors (TO-220 style, and the like), without simply connecting them in series-parallel (which would worsen the frequency response). Now that I've shown the equi-power network exists, it should be quite achievable to do 200-1000W with these parts, with good performance into the 100s of MHz. Which is far more than anything I have on my project list at the moment, so I'm happy. :)

Tim

--
Seven Transistor Labs, LLC 
Electrical Engineering Consultation and Contract Design 
Website: http://seventransistorlabs.com

Right. The structure is straightforward to analyze, though, because you can take advantage of self-similarity - the net admittance looking into the infinite ladder is the same as the parallel admittance of the same infinite ladder plus one more stage. So you have an implicit equation for the total admittance of the network, you know how the current divides up between the "one more" stage and the rest of then network for a given set of resistor values, and while I haven't worked the math on it it's probably possible to derive a closed-form sequence of values for the series and parallel resistors such that each one burns equal power.

And yeah the values do approach 0 and infinity as a limit, but the current goes to 0 in the limit as well, so as long as the resistor sequence meets some kind of standard for convergence (monotonically increasing rather than decreasing values for a start) you end up with an infinite network that burns finite power.

I think you even manage to avoid the thorny question that appears in the old "infinite 2D resistor lattice" problem of "Where does the current at infinity go to?" Which also sort of has implications for the whole universe, as if our universe is topologically infinite some cosmological theories shows that means there's a white-hole like singularity "at infinity" that any amount of anything can exit and enter through. So the laws of thermodynamics can't hold at a macro scale in a universe like that. OoooOooo...

Yeah, if you get a 5th order or greater polynomial in your system of equations you're f***ed. ;-)

Yep.

That doesn't follow at all. There are bazillions (technical term) of for example indefinite integrals that we know have antiderivatives but cannot be expressed in terms of elementary functions, and bazillions of definite integrals that we know converge but cannot be expressed in closed-form.

Some can be evaluated in terms of power series or "special functions" or whatever, but at the end of the day a power series solution around a particular point is often not terribly much more useful or enlightening than a numerical solution.

I'm a design engineer, not a scientist, not a manager. I hire people to manage my company and manage me. If I need serious theory, I know people who can do it for me.

I am very Irish and practical, so I prefer numerical solutions to abstract ones, and experiment to theory. I build things.

I respect physics and thermo, in the sense that I realize that I can't violate first principles and shouldn't waste my time trying. But I also don't allow tradition or restricted visions of theory to prevent me from considering new ideas for products.

Yes; but within reason and attention span. As Facebook says, done is better than perfect.

Causality and conservation of energy seem to matter. LC filters involve both.

I don't have pretensions to scientific knowledge. I passed the usual college courses in physics and chemistry and thermo and materials and electromagnetics, and they mostly trained my instincts as opposed to encouraging me to cover pages with equations.

The VAT20 is 1/2 watt, only moderate bandwidth. It's a cheap attenuator, the kind you buy a bunch of for the bench.

An attenuator, like that HP thing, need not have an equivalent lumped schematic.

Likely.

I've long been intrigued by those giant (like cubic foot) GHz KW heatsink-finned power attenuators and terminators. How can anything that big be fast?

You can buy small AlN-substrate terminators and attenuators, rated for hundreds of watts if you can heat sink them somehow.

Have you looked at the Caddock axial super-wideband resistors? They are handy for making several-kilovolt several-GHz pulse attenuators.

--

John Larkin         Highland Technology, Inc 

lunatic fringe electronics

Digikey, to be exact.

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John Larkin         Highland Technology, Inc 

lunatic fringe electronics

The ultimate wideband power attenuator might be a sheet of resistance material over a substrate, a distributed-lossy microstrip structure.

--

John Larkin         Highland Technology, Inc 

lunatic fringe electronics

Ahh, OK I was wondering what the goal was... Why is your network better than than just a string of ~10 20W TO-220's? (With some other R's (in the Tee) to keep 50 ohms everywhere.) I don't know much about R's at high frequency.

George H.

The low-order equivalent circuit of a TO-220 resistor will be two series inductors, representing lead (and bond, if applicable) inductance, a resistance between them, and a distributed capacitance from the resistor to the heatsink tab and other surroundings.

Really, the leads will have a transmission line impedance on the order of

150 ohms, which looks inductive at lower frequencies and lower impedances (say, relative to 50 ohms). The resistive patch will have some transmission line characteristic as well, and if it evenly covers the substrate, it will have a low impedance, and therefore look capacitive at lower frequencies.

"Lower frequencies" being, those well below the equivalent electrical length of the elements in question. So, for a TO-220, a couple cm -- a couple GHz.

At higher frequencies, you cannot so easily ignore the transmission line effects, and you have to build more and more complicated (higher order) lumped models (distributed L and C) to suit (or use transmission lines proper).

But at that point, you'll have so much trouble matching to these structures, and in keeping them consistent between assemblies (because we're talking lead geometry here), that you're better off giving up and declaring that as your upper frequency limit...

Tim

--
Seven Transistor Labs, LLC 
Electrical Engineering Consultation and Contract Design 
Website: http://seventransistorlabs.com

Or even better, one with tapered resistivity, so it's got lower losses where there's the most power, and so on.

A spool of RG-58 is a pretty good resistor, if a nonuniform one. It's been done before.

After a hundred feet or so, you wouldn't even much care about a wirewound resistor as the final termination!

Tim

--
Seven Transistor Labs, LLC 
Electrical Engineering Consultation and Contract Design 
Website: http://seventransistorlabs.com

You could short it after it accumulates 50 ohms of resistance.

--

John Larkin         Highland Technology, Inc 

lunatic fringe electronics

Say! Need some RG-58 or RG-59 male crimp-on connectors by any chance? I seem to have acquired a ton of them...

What happened to Vout?

I'd use a spread sheet for this. Spread sheets are pretty functional and easy to do.

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Rick C

His private stock. He only needs 12.

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Rick C

That would give the solution away to the first TDR anyone applies: better to use a 50 ohm load, even if it's nonideal.

I've always been rather fond of those controlled-impedance attenuators, that just have a half dozen '3dB' switches.